Do you apologize even if it's NOT your fault?

A lot of people have a hard time just saying the word "sorry" even if it's obviously their fault. Now, imagine if it wasn't your fault, would you be able to say "sorry"? Would you apologize for something even if it meant swallowing your pride? Or would you let err... "nature" run its course?
People who do apologize say that they do so because they just want the "fight" to be over and done with. Not apologizing would drag the fight longer. Would you apologize even if it's not your fault? What would be your reasons for apologizing or not apologizing? Do tell.